STATISTICAL BIAS IN DISTANCE AND PECULIAR VELOCITY ESTIMATION .1. THECALIBRATION-PROBLEM

This paper is the first in a series which treats comprehensively stati stical biases that affect the analysis of galaxy distance indicator da ta. The discussion will be specialized to the case of the Tully-Fisher relation for spiral galaxies. Statistical biases arising in distance indicator studies fall into two general categories. Those in the first category, the topic of this paper, occur when the data analysis is co nditioned upon assumptions about galaxy distances which are unrelated to the distance indicator information. The archetypical examples of su ch an approach are when a sample of galaxies is assigned a common dist ance based on assumed cluster membership, and when a sample is assumed to follow a well-defined redshift-distance relation. For reasons whic h are elaborated in the main body of the paper, the resultant statisti cal biases are described as comprising the ''calibration problem.'' St atistical biases in the second category will be the topic of the secon d paper in this series. They occur when individual galaxy distances ar e inferred directly from a distance indicator, without regard to clust er membership or redshift information. This series will also emphasize the distinction between the statistical properties of the ''forward'' and ''inverse'' representations of distance indicator relations. In t he case of Tully-Fisher, the former depicts the relation as a predicto r of absolute magnitude, given a value of rotation velocity; the latte r characterizes the relation as a predictor of rotation velocity given absolute magnitude. The use of the inverse method has been seen by so me workers as a panacea for bias effects. A goal of this series is to show that in general the inverse method exhibits biases analogous to, though different in detail from, those exhibited by the forward relati on. The form of calibration biases is determined by sample selection c riteria. The main results of this paper are accurate expressions for t he calibration biases which arise under several realistic types of sam ple selection, and a straightforward numerical technique to correct fo r such biases. The cases considered in detail are those in which sampl e selection is based upon photographic magnitude or diameter limits, w hile the Tully-Fisher photometry itself uses CCD or infrared magnitude s. Numerical simulations are presented to illustrate the bias effects and verify the efficacy of the correction procedure.